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Theorem tanatan 23838
Description: The arctangent function is an inverse to  tan. (Contributed by Mario Carneiro, 2-Apr-2015.)
Assertion
Ref Expression
tanatan  |-  ( A  e.  dom arctan  ->  ( tan `  (arctan `  A )
)  =  A )

Proof of Theorem tanatan
StepHypRef Expression
1 atancl 23800 . . 3  |-  ( A  e.  dom arctan  ->  (arctan `  A )  e.  CC )
2 2efiatan 23837 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1 ) )
32oveq1d 6303 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  +  1 ) )
4 2mulicn 10833 . . . . . . . 8  |-  ( 2  x.  _i )  e.  CC
54a1i 11 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  e.  CC )
6 atandm 23795 . . . . . . . . 9  |-  ( A  e.  dom arctan  <->  ( A  e.  CC  /\  A  =/=  -u _i  /\  A  =/= 
_i ) )
76simp1bi 1022 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  A  e.  CC )
8 ax-icn 9595 . . . . . . . 8  |-  _i  e.  CC
9 addcl 9618 . . . . . . . 8  |-  ( ( A  e.  CC  /\  _i  e.  CC )  -> 
( A  +  _i )  e.  CC )
107, 8, 9sylancl 667 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  e.  CC )
11 subneg 9920 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  _i  e.  CC )  -> 
( A  -  -u _i )  =  ( A  +  _i ) )
127, 8, 11sylancl 667 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( A  -  -u _i )  =  ( A  +  _i ) )
136simp2bi 1023 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  A  =/=  -u _i )
148negcli 9939 . . . . . . . . . 10  |-  -u _i  e.  CC
15 subeq0 9897 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =  0  <->  A  =  -u _i ) )
1615necon3bid 2667 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  -u _i  e.  CC )  ->  ( ( A  -  -u _i )  =/=  0  <->  A  =/=  -u _i ) )
177, 14, 16sylancl 667 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( ( A  -  -u _i )  =/=  0  <->  A  =/=  -u _i ) )
1813, 17mpbird 236 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( A  -  -u _i )  =/=  0 )
1912, 18eqnetrrd 2691 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( A  +  _i )  =/=  0 )
205, 10, 19divcld 10380 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  /  ( A  +  _i ) )  e.  CC )
21 ax-1cn 9594 . . . . . 6  |-  1  e.  CC
22 npcan 9881 . . . . . 6  |-  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  1 )  +  1 )  =  ( ( 2  x.  _i )  / 
( A  +  _i ) ) )
2320, 21, 22sylancl 667 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  +  1 )  =  ( ( 2  x.  _i )  /  ( A  +  _i )
) )
243, 23eqtrd 2484 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =  ( ( 2  x.  _i )  /  ( A  +  _i ) ) )
25 2muline0 10834 . . . . . 6  |-  ( 2  x.  _i )  =/=  0
2625a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  ( 2  x.  _i )  =/=  0 )
275, 10, 26, 19divne0d 10396 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  /  ( A  +  _i ) )  =/=  0
)
2824, 27eqnetrd 2690 . . 3  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =/=  0 )
29 tanval3 14181 . . 3  |-  ( ( (arctan `  A )  e.  CC  /\  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  +  1 )  =/=  0 )  ->  ( tan `  (arctan `  A ) )  =  ( ( ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) ) )
301, 28, 29syl2anc 666 . 2  |-  ( A  e.  dom arctan  ->  ( tan `  (arctan `  A )
)  =  ( ( ( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) ) )
312oveq1d 6303 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  -  1 )  =  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  -  1 ) )
3221a1i 11 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  1  e.  CC )
3320, 32, 32subsub4d 10014 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  -  1 )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  (
1  +  1 ) ) )
34 df-2 10665 . . . . . . . 8  |-  2  =  ( 1  +  1 )
3534oveq2i 6299 . . . . . . 7  |-  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  (
1  +  1 ) )
3633, 35syl6eqr 2502 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  1 )  -  1 )  =  ( ( ( 2  x.  _i )  / 
( A  +  _i ) )  -  2 ) )
3731, 36eqtrd 2484 . . . . 5  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  -  1 )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 ) )
38 2cn 10677 . . . . . . . 8  |-  2  e.  CC
39 mulcl 9620 . . . . . . . 8  |-  ( ( 2  e.  CC  /\  ( A  +  _i )  e.  CC )  ->  ( 2  x.  ( A  +  _i )
)  e.  CC )
4038, 10, 39sylancr 668 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( A  +  _i ) )  e.  CC )
415, 40, 10, 19divsubdird 10419 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  -  ( 2  x.  ( A  +  _i ) ) )  / 
( A  +  _i ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i )
)  -  ( ( 2  x.  ( A  +  _i ) )  /  ( A  +  _i ) ) ) )
42 mulneg12 10054 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( -u 2  x.  A )  =  ( 2  x.  -u A
) )
4338, 7, 42sylancr 668 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( -u
2  x.  A )  =  ( 2  x.  -u A ) )
44 negsub 9919 . . . . . . . . . . . 12  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  +  -u A )  =  ( _i  -  A ) )
458, 7, 44sylancr 668 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  ( _i  +  -u A )  =  ( _i  -  A
) )
4645oveq1d 6303 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( ( _i  +  -u A
)  -  _i )  =  ( ( _i 
-  A )  -  _i ) )
477negcld 9970 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  -u A  e.  CC )
48 pncan2 9879 . . . . . . . . . . 11  |-  ( ( _i  e.  CC  /\  -u A  e.  CC )  ->  ( ( _i  +  -u A )  -  _i )  =  -u A
)
498, 47, 48sylancr 668 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( ( _i  +  -u A
)  -  _i )  =  -u A )
508a1i 11 . . . . . . . . . . 11  |-  ( A  e.  dom arctan  ->  _i  e.  CC )
5150, 7, 50subsub4d 10014 . . . . . . . . . 10  |-  ( A  e.  dom arctan  ->  ( ( _i  -  A )  -  _i )  =  ( _i  -  ( A  +  _i )
) )
5246, 49, 513eqtr3rd 2493 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  ( _i 
-  ( A  +  _i ) )  =  -u A )
5352oveq2d 6304 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  -  ( A  +  _i ) ) )  =  ( 2  x.  -u A
) )
5438a1i 11 . . . . . . . . 9  |-  ( A  e.  dom arctan  ->  2  e.  CC )
5554, 50, 10subdid 10071 . . . . . . . 8  |-  ( A  e.  dom arctan  ->  ( 2  x.  ( _i  -  ( A  +  _i ) ) )  =  ( ( 2  x.  _i )  -  (
2  x.  ( A  +  _i ) ) ) )
5643, 53, 553eqtr2rd 2491 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  _i )  -  ( 2  x.  ( A  +  _i ) ) )  =  ( -u 2  x.  A ) )
5756oveq1d 6303 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  -  ( 2  x.  ( A  +  _i ) ) )  / 
( A  +  _i ) )  =  ( ( -u 2  x.  A )  /  ( A  +  _i )
) )
5854, 10, 19divcan4d 10386 . . . . . . 7  |-  ( A  e.  dom arctan  ->  ( ( 2  x.  ( A  +  _i ) )  /  ( A  +  _i ) )  =  2 )
5958oveq2d 6304 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  (
( 2  x.  ( A  +  _i )
)  /  ( A  +  _i ) ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 ) )
6041, 57, 593eqtr3d 2492 . . . . 5  |-  ( A  e.  dom arctan  ->  ( (
-u 2  x.  A
)  /  ( A  +  _i ) )  =  ( ( ( 2  x.  _i )  /  ( A  +  _i ) )  -  2 ) )
6137, 60eqtr4d 2487 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( exp `  ( 2  x.  ( _i  x.  (arctan `  A ) ) ) )  -  1 )  =  ( (
-u 2  x.  A
)  /  ( A  +  _i ) ) )
6224oveq2d 6304 . . . . 5  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  +  1 ) )  =  ( _i  x.  ( ( 2  x.  _i )  /  ( A  +  _i )
) ) )
638, 38, 8mul12i 9825 . . . . . . . 8  |-  ( _i  x.  ( 2  x.  _i ) )  =  ( 2  x.  (
_i  x.  _i )
)
64 ixi 10238 . . . . . . . . 9  |-  ( _i  x.  _i )  = 
-u 1
6564oveq2i 6299 . . . . . . . 8  |-  ( 2  x.  ( _i  x.  _i ) )  =  ( 2  x.  -u 1
)
6621negcli 9939 . . . . . . . . 9  |-  -u 1  e.  CC
6738mulm1i 10060 . . . . . . . . 9  |-  ( -u
1  x.  2 )  =  -u 2
6866, 38, 67mulcomli 9647 . . . . . . . 8  |-  ( 2  x.  -u 1 )  = 
-u 2
6963, 65, 683eqtri 2476 . . . . . . 7  |-  ( _i  x.  ( 2  x.  _i ) )  = 
-u 2
7069oveq1i 6298 . . . . . 6  |-  ( ( _i  x.  ( 2  x.  _i ) )  /  ( A  +  _i ) )  =  (
-u 2  /  ( A  +  _i )
)
7150, 5, 10, 19divassd 10415 . . . . . 6  |-  ( A  e.  dom arctan  ->  ( ( _i  x.  ( 2  x.  _i ) )  /  ( A  +  _i ) )  =  ( _i  x.  ( ( 2  x.  _i )  /  ( A  +  _i ) ) ) )
7270, 71syl5eqr 2498 . . . . 5  |-  ( A  e.  dom arctan  ->  ( -u
2  /  ( A  +  _i ) )  =  ( _i  x.  ( ( 2  x.  _i )  /  ( A  +  _i )
) ) )
7362, 72eqtr4d 2487 . . . 4  |-  ( A  e.  dom arctan  ->  ( _i  x.  ( ( exp `  ( 2  x.  (
_i  x.  (arctan `  A
) ) ) )  +  1 ) )  =  ( -u 2  /  ( A  +  _i ) ) )
7461, 73oveq12d 6306 . . 3  |-  ( A  e.  dom arctan  ->  ( ( ( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) )  =  ( ( ( -u 2  x.  A )  /  ( A  +  _i )
)  /  ( -u
2  /  ( A  +  _i ) ) ) )
7538negcli 9939 . . . . . 6  |-  -u 2  e.  CC
76 mulcl 9620 . . . . . 6  |-  ( (
-u 2  e.  CC  /\  A  e.  CC )  ->  ( -u 2  x.  A )  e.  CC )
7775, 7, 76sylancr 668 . . . . 5  |-  ( A  e.  dom arctan  ->  ( -u
2  x.  A )  e.  CC )
7875a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  -u 2  e.  CC )
79 2ne0 10699 . . . . . . 7  |-  2  =/=  0
8038, 79negne0i 9946 . . . . . 6  |-  -u 2  =/=  0
8180a1i 11 . . . . 5  |-  ( A  e.  dom arctan  ->  -u 2  =/=  0 )
8277, 78, 10, 81, 19divcan7d 10408 . . . 4  |-  ( A  e.  dom arctan  ->  ( ( ( -u 2  x.  A )  /  ( A  +  _i )
)  /  ( -u
2  /  ( A  +  _i ) ) )  =  ( (
-u 2  x.  A
)  /  -u 2
) )
837, 78, 81divcan3d 10385 . . . 4  |-  ( A  e.  dom arctan  ->  ( (
-u 2  x.  A
)  /  -u 2
)  =  A )
8482, 83eqtrd 2484 . . 3  |-  ( A  e.  dom arctan  ->  ( ( ( -u 2  x.  A )  /  ( A  +  _i )
)  /  ( -u
2  /  ( A  +  _i ) ) )  =  A )
8574, 84eqtrd 2484 . 2  |-  ( A  e.  dom arctan  ->  ( ( ( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  -  1 )  / 
( _i  x.  (
( exp `  (
2  x.  ( _i  x.  (arctan `  A
) ) ) )  +  1 ) ) )  =  A )
8630, 85eqtrd 2484 1  |-  ( A  e.  dom arctan  ->  ( tan `  (arctan `  A )
)  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886    =/= wne 2621   dom cdm 4833   ` cfv 5581  (class class class)co 6288   CCcc 9534   0cc0 9536   1c1 9537   _ici 9538    + caddc 9539    x. cmul 9541    - cmin 9857   -ucneg 9858    / cdiv 10266   2c2 10656   expce 14107   tanctan 14111  arctancatan 23783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614  ax-addf 9615  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-fi 7922  df-sup 7953  df-inf 7954  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-q 11262  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-ioo 11636  df-ioc 11637  df-ico 11638  df-icc 11639  df-fz 11782  df-fzo 11913  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13123  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-limsup 13519  df-clim 13545  df-rlim 13546  df-sum 13746  df-ef 14114  df-sin 14116  df-cos 14117  df-tan 14118  df-pi 14119  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-rest 15314  df-topn 15315  df-0g 15333  df-gsum 15334  df-topgen 15335  df-pt 15336  df-prds 15339  df-xrs 15393  df-qtop 15399  df-imas 15400  df-xps 15403  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-submnd 16576  df-mulg 16669  df-cntz 16964  df-cmn 17425  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-cnfld 18964  df-top 19914  df-bases 19915  df-topon 19916  df-topsp 19917  df-cld 20027  df-ntr 20028  df-cls 20029  df-nei 20107  df-lp 20145  df-perf 20146  df-cn 20236  df-cnp 20237  df-haus 20324  df-tx 20570  df-hmeo 20763  df-fil 20854  df-fm 20946  df-flim 20947  df-flf 20948  df-xms 21328  df-ms 21329  df-tms 21330  df-cncf 21903  df-limc 22814  df-dv 22815  df-log 23499  df-atan 23786
This theorem is referenced by:  atantanb  23843  atanord  23846
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